3.53.53 \(\int (1+e^{e^{\frac {1}{2} (20+x^2)}-x+x^2} (1-2 x-e^{\frac {1}{2} (20+x^2)} x)) \, dx\)

Optimal. Leaf size=25 \[ -1-e^{e^{10+\frac {x^2}{2}}-x+x^2}+x \]

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Rubi [A]  time = 0.15, antiderivative size = 24, normalized size of antiderivative = 0.96, number of steps used = 2, number of rules used = 1, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {6706} \begin {gather*} x-e^{x^2+e^{\frac {1}{2} \left (x^2+20\right )}-x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1 + E^(E^((20 + x^2)/2) - x + x^2)*(1 - 2*x - E^((20 + x^2)/2)*x),x]

[Out]

-E^(E^((20 + x^2)/2) - x + x^2) + x

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=x+\int e^{e^{\frac {1}{2} \left (20+x^2\right )}-x+x^2} \left (1-2 x-e^{\frac {1}{2} \left (20+x^2\right )} x\right ) \, dx\\ &=-e^{e^{\frac {1}{2} \left (20+x^2\right )}-x+x^2}+x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.22, size = 24, normalized size = 0.96 \begin {gather*} -e^{e^{10+\frac {x^2}{2}}-x+x^2}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1 + E^(E^((20 + x^2)/2) - x + x^2)*(1 - 2*x - E^((20 + x^2)/2)*x),x]

[Out]

-E^(E^(10 + x^2/2) - x + x^2) + x

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fricas [A]  time = 0.51, size = 20, normalized size = 0.80 \begin {gather*} x - e^{\left (x^{2} - x + e^{\left (\frac {1}{2} \, x^{2} + 10\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x*exp(1/4*x^2+5)^2+1-2*x)*exp(exp(1/4*x^2+5)^2+x^2-x)+1,x, algorithm="fricas")

[Out]

x - e^(x^2 - x + e^(1/2*x^2 + 10))

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giac [A]  time = 0.15, size = 20, normalized size = 0.80 \begin {gather*} x - e^{\left (x^{2} - x + e^{\left (\frac {1}{2} \, x^{2} + 10\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x*exp(1/4*x^2+5)^2+1-2*x)*exp(exp(1/4*x^2+5)^2+x^2-x)+1,x, algorithm="giac")

[Out]

x - e^(x^2 - x + e^(1/2*x^2 + 10))

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maple [A]  time = 0.10, size = 21, normalized size = 0.84

method	result	size
risch	\(x -{\mathrm e}^{{\mathrm e}^{\frac {x^{2}}{2}+10}+x^{2}-x}\)	\(21\)
default	\(x -{\mathrm e}^{{\mathrm e}^{\frac {x^{2}}{2}+10}+x^{2}-x}\)	\(23\)
norman	\(x -{\mathrm e}^{{\mathrm e}^{\frac {x^{2}}{2}+10}+x^{2}-x}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x*exp(1/4*x^2+5)^2+1-2*x)*exp(exp(1/4*x^2+5)^2+x^2-x)+1,x,method=_RETURNVERBOSE)

[Out]

x-exp(exp(1/2*x^2+10)+x^2-x)

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maxima [A]  time = 0.35, size = 20, normalized size = 0.80 \begin {gather*} x - e^{\left (x^{2} - x + e^{\left (\frac {1}{2} \, x^{2} + 10\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x*exp(1/4*x^2+5)^2+1-2*x)*exp(exp(1/4*x^2+5)^2+x^2-x)+1,x, algorithm="maxima")

[Out]

x - e^(x^2 - x + e^(1/2*x^2 + 10))

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mupad [B]  time = 0.10, size = 27, normalized size = 1.08 \begin {gather*} -{\mathrm {e}}^{-x}\,\left ({\mathrm {e}}^{x^2}\,{\mathrm {e}}^{{\mathrm {e}}^{10}\,{\mathrm {e}}^{\frac {x^2}{2}}}-x\,{\mathrm {e}}^x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1 - exp(exp(x^2/2 + 10) - x + x^2)*(2*x + x*exp(x^2/2 + 10) - 1),x)

[Out]

-exp(-x)*(exp(x^2)*exp(exp(10)*exp(x^2/2)) - x*exp(x))

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sympy [A]  time = 0.17, size = 15, normalized size = 0.60 \begin {gather*} x - e^{x^{2} - x + e^{\frac {x^{2}}{2} + 10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x*exp(1/4*x**2+5)**2+1-2*x)*exp(exp(1/4*x**2+5)**2+x**2-x)+1,x)

[Out]

x - exp(x**2 - x + exp(x**2/2 + 10))

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